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A097388 2n-th derivative of the Gaussian exp(-x^2) evaluated at x=0.

%I A097388 #44 Jan 22 2025 08:52:40
%S A097388 1,-2,12,-120,1680,-30240,665280,-17297280,518918400,-17643225600,
%T A097388 670442572800,-28158588057600,1295295050649600,-64764752532480000,
%U A097388 3497296636753920000,-202843204931727360000,12576278705767096320000,-830034394580628357120000,58102407620643984998400000
%N A097388 2n-th derivative of the Gaussian exp(-x^2) evaluated at x=0.
%C A097388 H_n(0) := n-th derivative of exp(-x^2) at x=0. H_(2*n + 1)(0) = 0 if n >= 0. - _Michael Somos_, Jan 24 2014
%C A097388 Self-convolution gives A076729(n)*(-1)^n. - _Vladimir Reshetnikov_, Oct 11 2016
%F A097388 E.g.f.: Sum_{k>=0} a(k) * x^(2*k) / (2*k)! = exp(-x^2).
%F A097388 a(n) = (-1)^n *(2*n)! / n!.
%F A097388 G.f.: 1/U(0) where U(k) = 1 + x*(2*k+2)/U(k+1); (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Nov 14 2012
%F A097388 G.f.: 1/Q(0), where Q(k) = 1 - x*(8*k+2) + x*(8*k+4)/(1 - x*(8*k+6) + x*(8*k+8)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, May 18 2013
%F A097388 a(n) = (-1)^n * A001813(n).
%F A097388 G.f. = 1 / (1 + 2*x / (1 + 4*x / (1 + 6*x / ... ))). - _Michael Somos_, Jan 24 2014
%F A097388 a(n) ~ (-1)^n*2^(2*n+1/2)*n^n/exp(n). - _Ilya Gutkovskiy_, Oct 11 2016
%F A097388 Sum_{n>=0} 1/a(n) = 1 - sqrt(Pi)*erfi(1/2)/(2*exp(1/4)). - _Amiram Eldar_, Nov 12 2020
%F A097388 From _Nikolaos Pantelidis_, Jan 08 2021: (Start)
%F A097388 G.f.: 1/G(1), where G(n) = 1+(8*n-6)*x-8*n*(2*n-1)*x^2/G(n+1); (Jacobi continued fraction)
%F A097388 G.f.: 1/(1 + 2*x - 8*x^2/(1 + 10*x - 48*x^2/(1 + 18*x - 120*x^2/(1 + 26*x - 224*x^2/(1 + 34*x - 360*x^2/(1+ 42*x - 448*x^2/(1+ 50*x - 648*x^2/( ...)))))))) (Jacobi continued fraction). (End)
%F A097388 a(n) = (-2)^n*n!*JacobiP(n, -1/2, -n, 3). - _Peter Luschny_, Jan 22 2025
%e A097388 exp(-x^2) = 1 - x^2 + x^4/4 - x^6/6 + ..., (d/dx)^4 exp(-x^2) = 12 - 60x^2 + ... so a(2)=12.
%e A097388 G.f. = 1 - 2*x + 12*x^2 - 120*x^3 + 1680*x^4 - 30240*x^5 + 665280*x^6 + ...
%p A097388 A097388 := n -> (-2)^n*n!*JacobiP(n, -1/2, -n, 3):
%p A097388 seq(simplify( A097388(n)), n = 0..18);  # _Peter Luschny_, Jan 22 2025
%t A097388 a[ n_] := If[ n < 0, 0, HermiteH[ 2 n, 0]]; (* _Michael Somos_, Jan 24 2014 *)
%o A097388 (PARI) {a(n) = if( n<0, 0, (-1)^n * (2*n)! / n!)};
%Y A097388 Cf. A001813, A076729.
%K A097388 sign
%O A097388 0,2
%A A097388 _Michael Somos_, Aug 12 2004